Kohei Fujimoto (University of Colorado), Daniel J. Scheeres (University of Colorado)
Keywords: Astrodynamics
Abstract:
Space situational awareness (SSA) is known to be a data starved problem compared to traditional estimation problems in that observation gaps per object may span over days if not weeks. Therefore, consistent characterization of the uncertainty associated with these objects including non-linear effects is crucial in maintaining an accurate catalog of objects in Earth orbit. Simultaneously, the motion of satellites in Earth orbit is well-modeled in that it is particularly amenable to having their solution and their uncertainty described through analytic or semi-analytic techniques. Even when stronger non-gravitational perturbations such as solar radiation pressure and atmospheric drag are encountered, these perturbations generally have deterministic components that are substantially larger than their time-varying stochastic components. Analytic techniques are powerful because time propagation is only a matter of changing the time parameter, allowing for rapid computational turnaround. These two ideas are combined in this paper: a method of analytically propagating non-linear orbit uncertainties is discussed. In particular, the uncertainty is expressed as an analytic probability density function (pdf) for all time. For a deterministic system model, such pdfs may be obtained if the initial pdf and the system states for all time are also given analytically. Even when closed-form solutions are not available, approximate solutions exist in the form of Edgeworth series for pdfs and Taylor series for the states. The coefficients of the latter expansion are referred to as state transition tensors (STTs), which are a generalization of state transition matrices to arbitrary order. Analytically expressed pdfs can be incorporated in many practical tasks in SSA. One can compute the mean and covariance of the uncertainty, for example, with the moments of the initial pdf as inputs. This process does not involve any sampling and its accuracy can be determined a priori. Analytical pdfs also enable the use of quadrature techniques in a Bayesian estimator for non-linear orbit determination and conjunction assessment. Finally, model parameter uncertainty, such as those for atmospheric drag, is readily implemented with little additional computational burden by adding the parameters in the state vector. To exemplify the speed and practicality of analytical techniques, we present two examples. First, numerically propagated Monte Carlo (MC) sample points from an initial Gaussian distribution are compared against the analytically propagated 3-sigma bounds of said distribution. The numerical simulation includes the J2 zonal harmonics of the Earth gravity field and atmospheric drag based on the NRLMSISE-00 density model. Conversely, the analytical model incorporates secular J2 effects and King-Heles solution for objects influenced by drag. We find that the MC results match the analytically propagated bounds well, which is expected as probability must be constant over any given volume for deterministic systems. Next, the mean and covariance of an initial Gaussian distribution are propagated with, again, both MC and analytical methods. Not only are the analytical results a good approximation of the MC, the computation time is reduced by several orders of magnitude.
Date of Conference: September 11-14, 2012
Track: Astrodynamics